The MSN \((\hat{X}, \hat{{A}}, \hat{{N}})\)  is cccc as a coproduct
of cccc MSNs (this is a general fact, in any category, a coproduct
of coproducts is a coproduct,
see [1]}), and \(\mathbf {p}\)  is supremum
preserving, according to REF (B) and (D). Observe that each
\(Z \in {E}\)  is also a subset of \(\hat{X}\)  and we denote by
\(\hat{\iota }_Z \colon Z \rightarrow \hat{X}\)  the corresponding inclusion
map.
Let \((Y, {B}, {M})\)  be a cccc MSN and \(\mathbf {q}\colon (Y, {B},{M}) \rightarrow (X, {A}, {N})\)  be a supremum preserving morphism,
represented by \(q \in \mathbf {q}\) . For all \(Z \in {E}\) , call \(q_Z \mathrel {\mathop :}=\hat{\iota }_Z \circ (q \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, q^{-1}(Z)) \colon q^{-1}(Z) \rightarrow \hat{X}\) . Because \(\mathbf {q}\)  is supremum preserving, \(Y = q^{-1}(X)\)  is
an \({M}\) -essential supremum of the family \(\langle q^{-1}(Z)\rangle _{Z \in {E}}\) . The family \({E}\)  being almost disjointed and \(q\)  being
measurable, \(q^{-1}(Z) \cap q^{-1}(Z^{\prime }) = q^{-1}(Z \cap Z^{\prime }) \in {M}\)  for any distinct \(Z, Z^{\prime } \in {E}\) . As a result, the family
\(\langle q_Z \rangle _{Z \in {E}}\)  subordinated to \(\langle q^{-1}(Z) \rangle _{Z\in {E}}\)  is compatible. By Proposition REF , this family
has a gluing \(r \colon Y \rightarrow \hat{X}\)  and, by Lemma REF ,
\(r\)  is \([({B}, {M}), (\hat{{A}}, \hat{{N}})]\) -measurable and
supremum preserving. Call \(p \mathrel {\mathop :}=\coprod _{Z \in {E}}\iota _Z\) . For each \(Z \in {E}\) , we have
\(q^{-1}(Z) \cap \lbrace p \circ r \ne q\rbrace  \subseteq q^{-1}(Z) \cap \lbrace r \ne \hat{\iota }_Z \circ (q \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, q^{-1}(Z))\rbrace  \in {M}.\) 
The family \(\lbrace q^{-1}(Z) : Z \in {E}\rbrace \)  is \({M}\) -generating and
\((Y, {B}, {M})\)  is locally determined, so we conclude that \(\lbrace p\circ r \ne q\rbrace  \in {M}\) , that is, \(\mathbf {p}\circ \mathbf {r}= \mathbf {q}\) .
As for uniqueness, let \(\mathbf {r}\)  be any morphism such that \(\mathbf {p}\circ \mathbf {r}= \mathbf {q}\) , and call \(r \in \mathbf {r}\)  one of its representatives. Fix \(Z\in {E}\) . Observe that \(\hat{\iota }_Z (p(z)) = z\)  for all \(z \in Z\) . For \({M}\) -almost every \(x \in q^{-1}(Z)\) , we have \(p(r(x)) =q(x)\)  which implies that \(r(x) \in Z \subseteq \hat{X}\) . For such an
\(x\) , we find that \(r(x) = \hat{\iota }_Z (p(r(x))) =\hat{\iota }_Z(q(x)) = q_Z(x)\) . Hence, \(r\)  is a gluing of the
compatible family \(\langle q_Z \rangle _{Z \in {E}}\)  and we invoke the
uniqueness part of Proposition REF  to conclude.
